Scalar curvature and geometric variational problems

标量曲率和几何变分问题

• 批准号: 2303624
• 负责人: Chao Li
• 金额: $ 38.05万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2303624&HistoricalAwards=false
• 关键词: Scalar; curvature; geometric; variational; problems

Curvature describes local bending of a space, and is used to distinguish how two shapes are different. This project concerns a particular notion of curvature, called the scalar curvature. Scalar curvature determines the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. Curvature arises throughout the natural sciences, and particularly in general relativity, where scalar curvature is the Lagrangian density of the Einstein-Hilbert action. A natural yet deep question is to understand the effects of scalar curvature on the global properties of a manifold. In particular, the project will focus on geometric variational problems, including minimal surfaces and soap bubbles. Minimal surfaces arise as the mathematical model of a number of interfaces in nature. In general relativity, minimal surfaces occur as “apparent horizons” of black holes; soap films and capillary interfaces also provide examples of minimal surfaces. Some key questions include the existence, regularity and topology of minimal surfaces. These two aspects of the research are deeply related, and will advance our understanding of the shape of nature. The PI will integrate this research with a variety of knowledge-disseminating activities that include organizing seminars, conferences, mini-courses, and reading groups, and giving public lectures. The proposed research concerns a range of topics in differential geometry, geometric measure theory and partial differential equations. Particularly, the PI would like to focus on the following four main topics. The first topic is the investigation of the obstruction problem for manifolds with positive scalar curvature, including the well-known `K(pi, 1) conjecture’. The second topic is to further understand the geometric comparison theorem for scalar curvature lower bound using Riemannian polyhedra. Minimal surface and soap bubbles are key technical tools for such problems. The third topic is to further investigate the stable Bernstein problem for minimal surfaces in R^n. The fourth topic is to further study the moduli space of positive scalar curvature metrics on 3-manifolds (with or without boundary). Understanding such questions has potential applications in 4-dimensional general relativity and in topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

曲率描述了空间的局部弯曲，用于区分两个形状的不同之处，该项目涉及一种特殊的曲率概念，称为标量曲率，它决定了黎曼中小测地线球的体积。流形与欧几里得空间中标准球的流形的偏差贯穿整个自然科学，特别是在标量的广义相对论中。曲率是爱因斯坦-希尔伯特作用的拉格朗日密度，一个自然而深刻的问题是了解标量曲率对流形全局属性的影响，该项目将重点关注几何变分问题，包括最小曲面和肥皂。最小表面作为自然界中许多界面的数学模型而出现，黑洞的“视界”也提供了。一些关键问题包括最小曲面的存在性、规律性和拓扑结构，这两个方面的研究密切相关，并将促进我们对自然形状的理解。知识传播活动，包括组织研讨会、会议、迷你课程和阅读小组，以及举办公开讲座。拟议的研究涉及微分几何、几何测度论和偏微分方程等一系列主题。重点关注以下方面四个主要主题。第一个主题是研究具有正标量曲率的流形的阻塞问题，包括著名的“K(pi,1)猜想”。第二个主题是进一步理解标量曲率的几何比较定理。使用黎曼多面体的下界是解决此类问题的关键技术工具。第三个主题是进一步研究 R^n 中最小曲面的稳定伯恩斯坦问题。旨在进一步研究 3 流形（有边界或无边界）上的正标量曲率度量的模空间，了解此类问题在 4 维广义相对论和拓扑中具有潜在应用。该奖项反映了 NSF 的法定使命，并被认为是值得的。通过使用基金会的智力优势和更广泛的影响审查标准进行评估来获得支持。